Lawrence lashed home right-footed from just outside the area in the 49th minute to earn Wales only their fourth victory in nine qualifying games and put them clear in second spot in Group D with 17 points, a point behind leaders Serbia.
Should third-placed Ireland suffer a shock home defeat by Moldova later on Friday Wales would be guaranteed at least runners-up spot in the group and a possible playoff place as they bid to reach the World Cup for the first time since 1958.
Wales, who are without injured talisman Gareth Bale, face Ireland on Monday in what could be a playoff for second place.
Only the eight runners-up with the best records from the nine European groups will take part in the two-legged playoffs and Wales were ninth best going into the match in Tbilisi.
